Formulas for the Walsh coefficients of smooth functions and their application to bounds on the Walsh coefficients

TL;DR

This paper derives formulas and bounds for Walsh coefficients of smooth functions, enhancing understanding of their behavior in various function spaces and aiding in applications like approximation and analysis.

Contribution

It introduces explicit formulas for Walsh coefficients of smooth functions and provides bounds, extending analysis to functions in reproducing kernel Hilbert spaces.

Findings

Formulas for Walsh coefficients of functions in $C^eta[0,1]$.

Upper bounds on Walsh coefficients for smooth functions.

Analysis of Walsh coefficients in reproducing kernel Hilbert spaces.

Abstract

We establish formulas for the $b$-adic Walsh coefficients of functions in $C^\alpha[0,1]$ for an integer $\alpha \geq 1$ and give upper bounds on the Walsh coefficients of these functions. We also study the Walsh coefficients of periodic and non-periodic functions in reproducing kernel Hilbert spaces.